Start by melting 1 cup of unsalted butter in a light-colored saucepan over medium heat. Stir constantly for about 5-6 minutes until the butter turns a rich golden brown and develops a nutty aroma. Once achieved, remove it from the heat and allow it to cool to room temperature for approximately 30 minutes.

In a large mixing bowl, cream together ½ cup of the cooled brown butter and dark brown sugar until the mixture is fluffy and well combined. Add in the pumpkin puree (or egg) and vanilla extract, mixing until smooth.

In another bowl, sift together all-purpose flour, old-fashioned oats, baking soda, ground cinnamon, and fine salt. Gradually fold this dry mixture into your wet ingredients, stirring gently until just combined.

Using a 1 tablespoon cookie scoop, portion the dough into small balls, creating about 32 dough balls in total. Place them on lined baking sheets, ensuring they are spaced about 1.5 inches apart.

Preheat your oven to 350°F (175°C). Once heated, bake the cookies for 8-9 minutes, or until the edges are set and the centers appear slightly puffy. After baking, let them cool on the baking sheet for 10 minutes before transferring them to wire racks.

In a mixing bowl, beat together the remaining brown butter with cream cheese until the mixture is smooth and creamy. Gradually mix in powdered sugar and a dash of vanilla extract.

To create the sandwich cookies, pair up two similar-sized cookies. Pipe a generous amount of the cream cheese frosting on one cookie and gently press the other cookie on top.